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«Abstract. These notes discuss U. Bader and Y. Shalom’s “Normal subgroup Theorem” for lattices in products of locally compact groups. No ...»

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Abstract. These notes discuss U. Bader and Y. Shalom’s “Normal subgroup Theorem”

for lattices in products of locally compact groups. No originality is claimed.

1. The main result

Definition 1.1. A lcsc group G is called just non-compact if its closed normal subgroups

are co-compact.

For instance simple groups are just non-compacts.

Theorem A (Normal subgroup Theorem). Let G1 and G2 be two non-discrete lcsc groups with property (T). Let Γ be an irreducible lattice in G1 × G2. If both G1 and G2 are just non-compact, so is Γ.

2. Poisson boundaries and the Factor Theorem Consider a locally compact second countable group G. A measure µ ∈ Prob(G) is called admissible if it is absolutely continuous with respect to the Haar measure and if its support generates G as a semi-group.

Definition 2.1. Assume that G (Y, η) is a non-singular action. The measure η is called µ-stationary if we have η = µ ∗ η. Equivalently, for all function f ∈ L1 (Y, η), we have f (gx)dµ(g)dη(x).

X In this situation, we say that (Y, η) is a (G, µ)-space.

We now define a special (G, µ)-space, the Poisson boundary. It somehow encodes “the behavior at infinity” of the µ-random walk on G.

2.1. Concrete Poisson boundaries. Let (G, µ) be as above. Consider the Borel space Ω = GN. Endow it with the product measure P = µ⊗N so that (Ω, P ) can be viewed as the space of increments of a µ-random walk on G. With this picture, denote by T : Ω → Ω the map T ((g1, g2, g3,... )) = (g1 g2, g3, g4,... ), which should be thought as a map accelerating

time. Moreover, the group G acts on Ω in the following way:

g · (g1, g2,... ) = (gg1, g2,... ).

We would like to define the Poisson boundary (B, νB ) of (G, µ) in such a way that L∞ (B, νB ) L∞ (Ω, P )T ⊂ L∞ (Ω, P ).

Since the action of G and T commute to each other, this subalgebra would be G-invariant and so G would act on (B, νB ).

The main problem with this idea is that neither T, nor G acts on (Ω, P ). Namely, the measure P is not quasi-invariant... So we need to be a bit careful with our construction.

Denote by µ0 a probability measure on G which is equivalent to the Haar measure. Then we can endow the space Ω with the measure P0 = µ0 ⊗ µ⊗N. This measure is quasi-invariant


under both T and the G-action, and P is absolutely continuous w.r.t. P0. Hence one can consider a standard Borel space (B, ν0 ) such that L∞ (B, ν0 ) L∞ (Ω, P0 )T ⊂ L∞ (Ω, P0 ).

Moreover G acts on (B, ν0 ) in such a way that the corresponding action on L∞ (B, ν0 ) corresponds to the G-action on (Ω, P0 ) with the above identification. We can assume that B is a compact metrizable space and that the action of G is continuous.

This means that we have a G-equivariant and T -invariant map π : (Ω, P0 ) → (B, ν0 ) such that π∗ P0 = ν0. Denote by νB the measure on B defined by νB = π∗ P, which is absolutely continuous w.r.t. ν0.

Although the measure P is not quasi-invariant on Ω, the measure νB is quasi-invariant.

Proposition 2.2. The measure νB satisfies µ ∗ νB = νB. This implies that it is G-quasiinvariant, and that (B, νB ) is a (G, µ)-space called the Poisson boundary of (G, µ).

Proof. Take f ∈ C(B). We have

–  –  –

Let us note a consequence that justifies our initial approach on the Poisson boundary.

Corollary 2.3. We claim that L∞ (B, ν0 ) = L∞ (B, νB ). Equivalently the measure νB has the same support as ν0 (they are equivalent).

Proof. Denote by A ⊂ B the support of νB, and put p = 1A ∈ L∞ (B, ν0 ). Then p is Ginvariant since νB is quasi-invariant for the G-action. View p as a T -invariant function in L∞ (Ω, P0 ). Being G-invariant, it does not depend on the first variable of Ω = GN. Since it is T -invariant, it has to be constant. Hence p = 1.

Remark 2.4.

The above proposition shows that for any G-space (Y, η) and every map ϕ : (Ω, P0 ) → (Y, η) which is T -invariant and G-equivariant P0 -almost everywhere, there exists a G-map ψ : (B, νB ) → (Y, η) such that ϕ = ψ ◦ π.

–  –  –

Example 2.5.

Take (G, µ) as before and assume that N is a closed normal subgroup of G.

Then the push-forward measure bar µ on G/N is admissible and the (G/N, bar µ) Poisson boundary (B, ν ) is isomorphic to the space of N -ergodic components of the (G, µ) Poisson

boundary (B, νB ):

L∞ (B, ν ) L∞ (B, νB )N.

2.2. The Factor Theorem.

Theorem 2.6 (Factor Theorem).

Consider two lcsc groups G1 and G2, with admissible measures µ1 and µ2. Then µ = µ1 × µ2 is admissible on G = G1 × G2. Denote by (B, νB ) the Poisson boundary of (G, µ). Assume that Γ G = G1 × G2 is an irreducible lattice. Then (i) any Γ-invariant von Neumann subalgebra of L∞ (B, νB ) is G-invariant.

(ii) any G-invriant von Neumann subalgebra of L∞ (B, νB ) is of the form A1 ⊗ A2 where G1 acts only on A1, and G2 acts only on A2.

Let us show that this theorem implies Theorem A. We will need two facts on the Poisson boundary. The first fact is about amenability of the action G (B, νB ), which will be proved later. The second fact is the triviality of some Poisson boundary for any amenable group.

Proposition 2.7. The action G (B, νB ) is amenable, in the sense that for any continuous X on a compact space, there exists a G-equivariant borel map B → Prob(X).

action G Even if it is not clear from this definition, amenability of actions passes to subgroups.

Theorem 2.8 (Black box).

If the group G is amenable, then there exists an admissible measure µ on G such that the Poisson boundary associated to (G, µ) is trivial ( i.e. reduced to a point).

It is easily checked that the converse holds as well: For any non-amenable group G with an admissible measure µ, the Poisson boundary of (G, µ) is non-trivial. For instance this follows from the previous proposition.

Proof of Factor Theorem ⇒ Normal subgroup Theorem. Assume that G1 and G2 are just non-compact, non discrete lcsc groups with property (T). Take an irreducible lattice Γ G1 × G2 and a non-trivial normal subgroup N Γ. Since Γ/N has property (T), it suffices to show that Γ/N is amenable.

Step 1. For i = 1, 2, Gi /Hi is amenable, where Hi = pri (N ).

Note that Hi is a normal subgroup of Gi, because it is closed and normalized by the dense subgroup pri (Γ). Since Gi is just non-compact, either Hi = {1} or Hi is co-compact in Gi (meaning that Gi /Hi is compact, hence amenable). It suffices to show that the first possibility never occurs. Assume for instance that H1 = {1}. This means that N is included in {1} × G2, and hence N = {1} × H2 is a non-trivial closed normal subgroup in G2, so it is co-compact in G2.

Now note that Γ/N is an irreducible lattice inside G/N = G1 × G2 /H2. Since G2 /H2 is compact, the projection pr1 (Γ/N ) is discrete in G1. The irreducibility condition then forces G1 to be discrete, which was excluded by our assumptions.

Step 2. Choosing a boundary.

We use the black box stated above and Step 1 to find, for i = 1, 2, a measure µi on Gi such that the push-forward measure bar µi on Gi /Hi has a trivial Poisson boundary. Denote by µ = µ1 × µ2 and by (B, νB ) the Poisson boundary associated with (G, µ).

Step 3. The proof.


Consider a continuous action Γ/N X on a compact space. We want to find an invariant measure on X. View X as a Γ space on which N acts trivially. Using amenability of the Poisson boundary, there exists a Γ-equivariant map ψ : B → Prob(X). Endow Y = Prob(X) with the measure η obtained by pushing forward νB. Then L∞ (Y, η) embeds into L∞ (B, νB ) as a Γ-invariant von Neumann sublagebra. Applying the Factor theorem, we get that it is in fact G-invariant and it is of the form A1 ⊗ A2, where G1 acts only on A1 and G2 acts on A2. Then Ai is Hi -invariant.

This shows that A1 is contained in L∞ (B, νB )H1 ×G2. But by the above example, this algebra is isomorphic to the L∞ algebra of the Poisson boundary associated with G/(H1 × G2 ) G1 /H1 and the measure bar µ1. By our choice of µ1, this algebra is trivial and A1 C.

Similarly A2 C, thus showing that the measure η on Y is a Dirac measure. The corresponding point is then a Γ-invariant point in Y, i.e. a Γ-invariant measure on X.

3. More on Poisson boundaries

In this section, we consider a lcdc group G with an admissible measure µ ∈ Prob(G). We will define Ω = GN, endowed with the probability measure P0 = µ0 ⊗ µ⊗N, where µ0 is equivalent to the Haar measure.

Recall that (B, ν0 ) is a factor of (Ω, P0 ), with a G-equivariant, T -invariant factor map π and that the stationary measure νB ∈ Prob(B) is the push-forward measure on B of the measure P = µ⊗N ∈ Prob(Ω). Recall that νB is equivalent to ν0.

3.1. Poisson boundaries and Harmonic functions. We relate here Harmonic functions on G and functions on the Poisson boundary.

Definition 3.1. A function φ ∈ L∞ (G, Haar) is said µ-harmonic (on the right) if φ = φ ∗ µ, in other words φ(s) = G φ(sg)dµ(g) for almost every s ∈ G. Denote by Har(µ) the set of µ-harmonic functions on G.

Proposition 3.2. Take φ ∈ Har(µ). Then for P0 -almost all ω = (g1, g2,... ) ∈ Ω the sequence φ(g1 g2 · · · gn ) converges to a number quantity by bar φ(ω). Note that bar φ is T invariant so it naturally defines a function on B by Remark 2.4. Moreover Φ → bar φ is a G-equivariant map.

Proof. Denote by Xn : Ω → R the random variable defined by Xn (ω)φ(g1 g2 · · · gn ). By harmonicity of φ, we see that for all n ≥ 1, Xn is the conditional expectation of Xn+1 on the subalgebra An := L∞ (G, µ0 ) ⊗ L∞ (G, µ) ⊗ n−1. In other words, Xn define a (bounded) martingale for this increasing sequence of subalgebras. Hence the result follows from the martingale convergence theorem.

Definition 3.3. For a (G, µ)-space (Y, η) denote by Pη : L∞ (Y, η) → Har(µ) the Poisson transform, defined by the formula Pη (f )(g) = X f (gx)dη(x), for all f ∈ L∞ (Y, η), G ∈ G.

We leave it as an exercise that Pη (f ) is indeed a harmonic function.

Theorem 3.4.

The G-maps φ ∈ Har(µ) → bar φ ∈ L∞ (B, νB ) and PνB : L∞ (B, νB ) → Har(µ) are inverse of each other.

Proof. First, take φ ∈ Har(µ). By definition of νB, we have

–  –  –

3.2. conditional measures and (G, µ)-boundaries.

Lemma 3.6.

Assume that Y is a compact metric space on which G acts continuously and that η ∈ Prob(Y ) is a µ-stationary measure. Then there exists a (P0 -almost everywhere) G-equivariant, T -invariant map ω ∈ Ω → ηω ∈ Prob(Y ) such that Ω ηω dP (ω) = η, and for almost every ω = (g1, g2,... ) ∈ Ω, we have ηω = limn (g1 g2 · · · gn )η.

Because of Remark 2.4, the lemma implies that there exists a G-equivariant map b = π(ω) ∈ B → ηb := ηω ∈ Prob(Y ) such that B ηb dνB (b) = η.

Proof. Denote by C0 a countable dense subset in C(Y ), and denote by Ω0 the set of ω = (g1, g2,... ) ∈ Ω such that the limit limn Y f (g1 g2 · · · gn y)dη(y) = Pη (f )(g1 g2 · · · gn ) exists for all f ∈ C0. Since Pη (f ) is harmonic for all f ∈ C0 and C0 is countable, Proposition 3.2 implies that Ω0 is P0 -conul.

By density of C0, we get that for every ω = (g1, g2,... ) ∈ Ω0 and every f ∈ C(Y ), the sequence ( Y f (g1 g2 · · · gn y)dη(y))n converges to a limit ηω (f ). Obviously this is also true for ω ∈ G · Ω0, so we can assume that Ω0 is G-invariant. Moreover, the map f → ηω (f ) is a positive linear functional, i.e. ηω is a measure on Y. Clearly ω ∈ Ω0 → ηω ∈ Prob(Y ) is measurable and G-equivariant and we have for all f ∈ C(Y )

–  –  –

Proof. Assume that G Y is a continuous action on a compact metric space. We want to construct a G-equivariant measurable map B → Prob(Y ). By the above lemma, it is sufficient to find a µ-stationary measure η ∈ Prob(Y ). But a stationary measure is nothing but a fixed point for the map T : Prob(Y ) → Prob(Y ) defined by T η = µ ∗ η. Such a fixed point exists by Markov-Kakutani fixed point theorem: just pick any measure η0 ∈ Prob(Y ) and define η to be a weak*-limit point of ( N T k (η0 ))/N. This is a stationary measure.

k=1 Definition 3.8. The measures ηω from Lemma 3.6 are called the conditional measures. They depend a priori on a compact model for the action G (Y, η).


Definition 3.9. A (G, µ) space (Y, η) is called a (G, µ)-boundary if it is a factor of the Poisson boundary (B, νB ) i.e. there exists a G-equivariant measurable map ψ : B → Y such that ψ∗ νB = η.

Theorem 3.10.

Consider a (G, µ)-space (Y, η). The following are equivalent (1) (Y, η) is a (G, µ)-boundary;

(2) There exists a compact model such that the conditional measures are Dirac measures;

(3) For every compact model, the conditional measures are Dirac measures.

Moreover, if this happens, the factor map (B, νB ) → (Y, η) is essentially unique.

Proof. (1) ⇒ (3). Take a continuous compact model G Y0 for the action G (Y, η) and denote again by η the measure obtained on Y0 after identifying Y and Y0. Assume that there is a G-equivariant map ψ : (B, µ) → (Y0, η) such that ψ∗ νB = η. This last condition clearly implies a relation between the Poisson transforms: Pη (f ) = PνB (f ◦ ψ) for all f ∈ L∞ (Y0, η).

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